Twenty persons want to buy a $10 ticket each. Ten of them have a $10
note and others have a $20 note. The person at the ticket counter has
no money to start with. What is the probability that the person at the
ticket counter will not have a change problem?

Player A begins by placing a checker in the lower left-hand corner
of a checkerboard (8 by 8 squares). Player B places a checker one
square to the right or one square up or one square diagonally up and
to the right of Player A's checker... Would you rather be Player A or
Player B?

A student notices that calculators compute nCr for r > n, and wonders
about the combination function's "true" form. Doctor Vogler explains how
to extend various formulas to different types of arguments, including
non-integer and complex numbers.

Each of four rows of coins has exactly one penny, one nickel, one
dime, and one quarter. No row, either horizontal, vertical, or
diagonal, has more than one coin of each kind. How are the coins
arranged?

I understand the formula n!/((n-r)!*r!) to compute the number of
combinations of r objects selected from n objects. Now, having
determined nCr, how can I calculate the number of resulting
combinations which contain t items? Can I calculate that directly at
the start without first finding all the initial combinations?

A combinatorics student wonders why all four-digit PINs with exactly three different
digits does not amount to choosing three digits out of 10 without repetition. Doctor
Douglas explains how 10*9_C_2 differs from 10_C_3.